GATE ME 2020 Set 2 — Question 47
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Fluid Mechanics → Fluid Properties & Statics → Buoyancy & Stability
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Question
A hollow spherical ball of radius floats in still water, with half of its volume submerged. Taking the density of water as , and the acceleration due to gravity as , the natural frequency of small oscillations of the ball, normal to the water surface is ________ radians/s (round off to 2 decimal places).
Correct answer
8.6 to 8.7
Solution
1.Equilibrium condition: At equilibrium, the weight of the ball equals the buoyancy force.
Mass of the ball .
2.Restoring force: For a small vertical displacement from the equilibrium position, the additional buoyancy force acts as a restoring force.
, where is the cross-sectional area at the water surface.For a sphere at half-submerged level, .
3.Equation of motion:
4.Natural frequency:
Given and :
.
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